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Theorem infunsdom 9238
Description: The union of two sets that are strictly dominated by the infinite set 𝑋 is also strictly dominated by 𝑋. (Contributed by Mario Carneiro, 3-May-2015.)
Assertion
Ref Expression
infunsdom (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)

Proof of Theorem infunsdom
StepHypRef Expression
1 sdomdom 8137 . . 3 (𝐴𝐵𝐴𝐵)
2 infunsdom1 9237 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝐵𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
32anass1rs 634 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐵𝑋) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
43adantlrl 699 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
51, 4sylan2 580 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
6 simpll 750 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝑋 ∈ dom card)
7 sdomdom 8137 . . . . . . 7 (𝐵𝑋𝐵𝑋)
87ad2antll 708 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵𝑋)
9 numdom 9061 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐵𝑋) → 𝐵 ∈ dom card)
106, 8, 9syl2anc 573 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐵 ∈ dom card)
11 sdomdom 8137 . . . . . . 7 (𝐴𝑋𝐴𝑋)
1211ad2antrl 707 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴𝑋)
13 numdom 9061 . . . . . 6 ((𝑋 ∈ dom card ∧ 𝐴𝑋) → 𝐴 ∈ dom card)
146, 12, 13syl2anc 573 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → 𝐴 ∈ dom card)
15 domtri2 9015 . . . . 5 ((𝐵 ∈ dom card ∧ 𝐴 ∈ dom card) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1610, 14, 15syl2anc 573 . . . 4 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐵𝐴 ↔ ¬ 𝐴𝐵))
1716biimpar 463 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → 𝐵𝐴)
18 uncom 3908 . . . . . 6 (𝐴𝐵) = (𝐵𝐴)
19 infunsdom1 9237 . . . . . 6 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐵𝐴) ≺ 𝑋)
2018, 19syl5eqbr 4821 . . . . 5 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐵𝐴𝐴𝑋)) → (𝐴𝐵) ≺ 𝑋)
2120anass1rs 634 . . . 4 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ 𝐴𝑋) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2221adantlrr 700 . . 3 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ 𝐵𝐴) → (𝐴𝐵) ≺ 𝑋)
2317, 22syldan 579 . 2 ((((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) ∧ ¬ 𝐴𝐵) → (𝐴𝐵) ≺ 𝑋)
245, 23pm2.61dan 814 1 (((𝑋 ∈ dom card ∧ ω ≼ 𝑋) ∧ (𝐴𝑋𝐵𝑋)) → (𝐴𝐵) ≺ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wcel 2145  cun 3721   class class class wbr 4786  dom cdm 5249  ωcom 7212  cdom 8107  csdm 8108  cardccrd 8961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-isom 6040  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-2o 7714  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-oi 8571  df-card 8965  df-cda 9192
This theorem is referenced by:  csdfil  21918
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